Construction of locally compact nearfields (Q1754391)

Let \(F\mid K\) be a finite, tamely ramified, abelian extension of local fields. Extending results of \textit{R. Rink} [Geom. Dedicata 20, 93--119 (1986; Zbl 0592.12016)] and \textit{K. U. Hanke} and \textit{H. Wähling} [J. Geom. 39, No. 1--2, 92--115 (1990; Zbl 0732.12010)], the author describes (and counts) all continuous homomorphisms of \(F^\times\) into \(\mathrm{Gal}(F\mid K)\) that are couplings in the sense of \textit{H. Karzel} [Arch. Math. 16, 247--256 (1965; Zbl 0131.01701)]. These couplings yield locally compact (totally disconnected) Dickson nearfields.